Homogeneous para-Kahler Einstein manifolds

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HOMOGENEOUS PARA-K ¨AHLER EINSTEIN MANIFOLDS D.V.ALEKSEEVSKY,C.MEDORI AND A.TOMASSINI Dedicated to E.B.Vinberg on the occasion of his 70th birthday Abstract.A para-K¨a hler manifold can be defined as a pseudo-Riemannian manifold (M,g )with a parallel skew-symmetric para-complex structures K ,i.e.a parallel field of skew-symmetric endomor-phisms with K 2=Id or,equivalently,as a symplectic manifold (M,ω)with a bi-Lagrangian structure L ±,i.e.two complementary integrable Lagrangian distributions.A homogeneous manifold M =G/H of a semisimple Lie group G admits an invariant para-K¨a hler structure (g,K )if and only if it is a covering of the adjoint orbit Ad G h of a semisimple element h.We give a description of all invariant para-K¨a hler structures (g,K )on a such homogeneous ing a para-complex analogue of basic formulas of K¨a hler geometry,we prove that any invariant para-complex structure K on M =G/H defines a unique para-K¨a hler Einstein structure (g,K )with given non-zero scalar curvature.An explicit formula for the Einstein metric g is given.A survey of recent results on para-complex geometry is included.Contents 1.Introduction.22.A survey on para-complex geometry.42.1.Para-complex structures.42.2.Para-hypercomplex (complex product)structures.62.3.Para-quaternionic structures.72.4.Almost para-Hermitian and para-Hermitian structures.82.5.Para-K¨a hler (bi-Lagrangian or Lagrangian 2-web)structures.11
2.6.Para-hyperK¨a hler (hypersymplectic)structures and para-
hyperK¨a hler structures with torsion (PHKT-structures).13
2.7.Para-quaternionic K¨a hler structures and para-quaternionic-
K¨a hler with torsion (PQKT)structures.14
2 D.V.ALEKSEEVSKY,C.MEDORI AND A.TOMASSINI
2.8.Para-CR structures and para-quaternionic CR structures
(para-3-Sasakian structures).16 3.Para-complex vector spaces.17 3.1.The algebra of para-complex numbers C.17 3.2.Para-complex structures on a real vector space.18
3.3.Para-Hermitian forms.19
4.Para-complex manifolds.19 4.1.Para-holomorphic coordinates.20
4.2.Para-complex differential forms.21
5.Para-K¨a hler manifolds.22 5.1.Para-K¨a hler structures and para-K¨a hler potential.22 5.2.Curvature tensor of a para-K¨a hler metric.24 5.3.The Ricci form in para-holomorphic coordinates.25
5.4.The canonical form of a para-complex manifold.27
6.Homogeneous para-K¨a hler manifolds.29 6.1.The Koszul formula for the canonical form.29 6.2.Invariant para-complex structures on a homogeneous manifold.31 6.3.Invariant para-K¨a hler structures on a homogeneous reductive
manifold.32 7.Homogeneous para-K¨a hler Einstein manifolds of a semisimple
group.33 7.1.Invariant para-K¨a hler structures on a homogeneous manifold.33 7.2.Fundamental gradations of a real semisimple Lie algebra.34 putation of the Koszul form and the main theorem.36 7.4.Examples.38 References40
1.Introduction.
An almost para-complex structure on a2n-dimensional manifold M is afield K of involutive endomorphisms(K2=1)with n-dimensional eigendistributions T±with eigenvalues±1.(More generally,anyfield K of involutive endomorphisms is called an almost para-complex struc-ture in weak sense).
If the n-dimensional eigendistributions T±of K are involutive,thefield K is called a para-complex structure.This is equivalent to the vanishing of the Nijenhuis tensor N K of the K.In other words,a para-complex struc-ture on M is the same as a pair of complementary n-dimensional integrable distributions T±M.
A decomposition M=M+×M−of a manifold M into a direct product defines on M a para-complex structure K in the weak sense with eigendis-tributions T+=T M+and T−=T M−tangent to the factors.It is a para-complex structure in the factors M±have the same dimension.
Any para-complex structure locally can be obtained by this construction.
HOMOGENEOUS PARA-K¨AHLER EINSTEIN MANIFOLDS3 Due to this,an(almost)para-complex structure is also called an(almost) product structure.
A manifold M endowed with a para-complex structure K admits an atlas of para-holomorphic coordinates(which are functions with values in the alge-bra C=R+e R,e2=1,of para-complex numbers)such that the transition functions are para-holomorphic.
One can define para-complex analogues of different composed geomet-ric structures which involve a complex structure(Hermitian,K¨a hler,nearly K¨a hler,special K¨a hler,hypercomplex,hyper-K¨a hler,quaternionic,quater-nionic K¨a hler structures,CR-structure etc.)changing a complex structure J to a para-complex structure K.Many results of the geometry of such structures remain valid in the para-complex case.On the other hand,some para-complex composed structures admit different interpretation as3-webs, bi-Lagrangian structure and so on.
The structure of this paper is the following.
In section2,we give a survey of known results about para-complex geometry. Sections3,4and5contain an elementary introduction to para-complex and para-K¨a hler geometry.
The bundleΛr(T∗M⊗C)of C-valued r-form is decomposed into a direct sum of(p,q)-formsΛp,q M and the exterior differential d is represented as a direct sum d=∂+¯∂.Moreover,a para-complex analogue of the Dolbeault Lemma holds(see[33]and subsection4.2).
A para-K¨a hler structure on a manifold M is a pair(g,K)where g is a pseudo-Riemannian metric and K is a parallel skew-symmetric para-complex structure.A pseudo-Riemannian2n-dimensional manifold(M,g)admits a para-K¨a hler structure(g,K)if and only if its holonomy group is a subgroup of GL n(R)⊂SO n,n⊂GL2n(R).
If(g,K)is a para-K¨a hler structure on M,thenω=g◦K is a symplec-tic structure and the±1-eigendistributions T±M of K are two integrable ω-Lagrangian distributions.Due to this,a para-K¨a hler structure can be identified with a bi-Lagrangian structure(ω,T±M)whereωis a symplectic structure and T±M are two integrable Lagrangian distributions.
In section4,we derive some formulas for the curvature and Ricci curvature of a para-K¨a hler structure(g,K)in terms of para-holomorphic coordinates. In particular,we show that,as in the K¨a hler case,the Ricci tensor S de-pends only on the determinant of the metric tensor gα¯βwritten in terms of para-holomorphic coordinates.
In sections5and6,we consider a homogeneous manifold(M=G/H,K,vol) with an invariant para-complex structure K and an invariant volume form vol.We establish a formula which expresses the pull-backπ∗ρto G of the Ricci formρ=S◦K of any invariant para-K¨a hler structure(g,K)as the differential of a left-invariant1-formψ,called the Koszul form.
In the last section,we use the important result by Z.Hou,S.Deng,S. Kaneyuki and K.Nishiyama(see[60],[61])stating that a homogeneous
4 D.V.ALEKSEEVSKY,C.MEDORI AND A.TOMASSINI
manifold M=G/H of a semisimple Lie group G admits an invariant para-K¨a hler structure if and only if it is a covering of the adjoint orbit Ad G h of a semisimple element h∈g=Lie(G).
We describe all invariant para-complex structures K on M=G/H in terms of fundamental gradations of the Lie algebra g with g0=h:=Lie(H)and we show that they are consistent with any invariant symplectic structureωon G/H such that(g=ω◦K,K)is an invariant para-K¨a hler structure.This gives a description of all invariant para-K¨a hler structures on homogeneous manifolds of a semisimple group G.An invariant para-complex structure on M=G/H defines an Anosovflow,but a theorem by Y.Benoist and F. Labourie shows that thisflow can not be push down to any smooth compact quotientΓ\G/H.We give a complete description of invariant para-K¨a hler-Einstein metrics on homogeneous manifolds of a semisimple Lie group and prove the following theorem.
Theorem1.1.Let M=G/H be a homogeneous manifold of a semisimple Lie group G which admits an invariant para-K¨a hler structure and K be an invariant para-complex structure on M.Then there exists a unique invariant symplectic structureρon M which is the push down of the differential dψof the canonical Koszul1-formψon G such that gλ,K:=λ−1ρ◦K is an invariant para-K¨a hler Einstein metric with Einstein constantλ=0and this construction gives all invariant para-K¨a hler-Einstein metrics on M.
2.A survey on para-complex geometry.
2.1.Para-complex structures.The notion of almost para-complex structure(or almost product structure)on a manifold was introduced by P.K.Raˇs evski˘ı[92]and P.Libermann[78],[79],where the problem of integrability had been also discussed.The paper[36]contains a survey of further results on para-complex structure with more then100references. Note that a para-complex structure K on a manifold M defines a new Lie algebra structure in the space X(M)of vectorfields given by
[X,Y]K:=[KX,Y]+[X,KY]−K[X,Y]
such that the map
(X(M),[.,.]K)→(X(M),[.,.]),X→KX
is a homomorphism of Lie algebras.
Moreover,K defines a new differential d K of the algebraΛ(M)of differential forms which is a derivation ofΛ(M)of degree one with d2K=0.It is given by
d K:={d,K}:=d◦ιK+ιK◦d
whereιK is the derivation associated with K of the supercommutative al-gebraΛ(M)of degree−1defined by the contraction.(Recall that the su-perbracket of two derivations is a derivation).
In[77],the authors define the notion of para-complex affine immersion
HOMOGENEOUS PARA-K¨AHLER EINSTEIN MANIFOLDS5 f:M→M′with transversal bundle N between para-complex manifolds (M,K),(M′,K′)equipped with torsion free para-complex connections∇,∇′and prove a theorem about the existence and the uniqueness of such an im-mersion of(M,K,∇)into the affine space M′=R2m+2n with the standard para-complex structure andflat connection.
In[96],the author defines the notion of para-tt∗-bundle over an(almost) para-complex2n-dimensional manifold(M,K)as a vector bundleπ:E→M with aflat connection∇and a End(E)-valued1-form S such that1-parametric family of connections
∇t X=cosh(t)∇X+sinh(t)∇KX,X∈T M
isflat.It is a para-complex version of tt∗-bundle over a complex manifold defined in the context of quantumfield theory in[31],[43]and studied from differential-geometric point of view in[35].In[96],[97],[98],the author studies properties of para-tt∗-connection∇t on the tangent bundle E=T M of an(almost)para-complex manifold M.In particular,he shows that nearly para-K¨a hler and special para-complex structures on M provide para-tt∗-connections.It is also proved that a para-tt∗-connection which preserves a metric or a symplectic form determines a para-pluriharmonic map f:M→N where N=Sp2n(R)/U n(C n)or,respectively,N=SO n,n/U n(C n)with invariant pseudo-Riemannian metric.Here U n(C n)stands for para-complex analogue of the unitary group.
Generalized para-complex structures.Let T(M):=T M⊕T∗M be the gener-alized tangent bundle of a manifold M i.e.the direct sum of the tangent and cotangent bundles equipped with the natural metric g of signature(n,n),
g (X,ξ),(X′,ξ′) :=1
d ξ′(X)−ξ(X′)
2
where L X is the Lie derivative in the direction of a vectorfield X.A max-imally g-isotropic subbundle D⊂T(M)is called a Dirac structure if its space of sectionsΓ(D)is closed under the Courant bracket.
Changing in the definition of the para-complex structure the tangent bundle T M to the generalized tangent bundle T(M)and the Nijenhuis bracket to the Courant bracket,A.Wade[102]and I.Vaisman[101]define the notion of a generalized para-complex structure which unifies the notion of symplec-tic structure,Poisson structure and para-complex structure and it is similar to the Hitchin’s definition of a generalized complex structure(see e.g[55],
[62]).
A generalized para-complex structure is afield K of involutive skew-symmetric endomorphisms of the bundle T(M)whose±1-eigendistributions T±are closed under the Courant bracket.
6 D.V.ALEKSEEVSKY,C.MEDORI AND A.TOMASSINI
In other words,it is a decomposition T(M)=T+⊕T−of the generalized tangent bundle into a direct sum of two Dirac structures T±. Generalized para-complex structures naturally appear in the context of mir-ror symmetry:a semi-flat generalized complex structure on a n-torus bundle with sections over an n-dimensional manifold M gives rise to a generalized para-complex structure on M,see[21].I.Vaisman[101]extends the reduc-tion theorem of Marsden-Weinstein type to generalized complex and para-complex structures and gives the characterization of the submanifolds that inherit an induced structure via the corresponding classical tensorfields. 2.2.Para-hypercomplex(complex product)structures.An(al-most)para-hypercomplex(or an almost complex product struc-ture)on a2n-dimensional manifold M is a pair(J,K)of an anticommuting (almost)complex structure J and an(almost)para-complex structure K. The product I=JK is another(almost)para-complex structure on M. If the structures J,K are integrable,then I=JK is also an(integrable) para-complex structure and the pair(J,K)or triple(I,J,K)is called a para-hypercomplex structure.An(almost)para-hypercomplex struc-ture is similar to an(almost)hypercomplex structure which is defined as a pair of anticommuting(almost)complex structures.Like for almost hyper-complex structure,there exists a canonical connection∇,called the Obata connection,which preserves a given almost para-hypercomplex structure (i.e.such that∇J=∇K=∇I=0).The torsion of this connection van-ishes if and only if N J=N K=N I=0that is(J,K)is a para-hypercomplex structure.
At any point x∈M the endomorphisms I,J,K define a standard basis of the Lie subalgebra sl2(R)⊂End(T x M).The conjugation by a(constant) matrix A∈SL2(R)allows to associate with an(almost)para-hypercomplex structure(J,K)a3-parametric family of(almost)para-hypercomplex struc-tures which have the same Obata connection.
Let T M=T+⊕T−be the eigenspace decomposition for the almost para-complex structure K.Then the almost complex structure J defines the isomorphism J:T+→T−and we can identify the tangent bundle T M with a tensor product T M=R2⊗E such that the endomorphisms J,K,I acts on thefirst factor R2in the standard way:
(1)J= 0−110 ,K= 100−1 ,I=JK= 0110 . Any basis of E x defines a basis of the tangent space T x M and the set of such(adapted)bases form a GL n(R)-structure(that is a principal GL n(R)-subbundle of the frame bundle of M).So one can identify an almost para-hypercomplex structure with a GL n(R)-structure and a para-hypercomplex structure with a1-integrable GL n(R)-structure.This means that(J,K)is a para-hypercomplex structure.The basic facts of the geometry of para-hypercomplex manifolds are described in[13],where also some examples are considered.
HOMOGENEOUS PARA-K¨AHLER EINSTEIN MANIFOLDS7 Invariant para-hypercomplex structures on Lie groups are investigated in[14],[17].Algebraically,the construction of left-invariant para-hypercomplex structures on a Lie group G reduces to the decomposition of its Lie algebra g into a direct sum of subalgebras g+,g−together with the construction of a complex structure J which interchanges g+,g−.It is proved in[12]that the Lie algebras g±carry a structure of left-symmetric algebra.Applications to construction of hypercomplex and hypersymplectic (or para-hyperK¨a hler)structures are considered there.
Note that a para-hypercomplex structure(J,K)on a2n-dimensional man-ifold M can be identified with a3-web,that is a triple(V1,V2,V3)of mu-tually complementary n-dimensional integrable distributions.Indeed,the eigendistributions T±,S±of the para-complex structures K and I=JK defines a3-web(T+,T−,S+).Conversely,let(V1,V2,V3)be a3-web.Then the decomposition T M=V1+V2defines a para-complex structure K and the distribution V3is the graph of a canonically defined isomorphism f:V1→V2 that is V3=(1+f)V1.The n-dimensional distribution V4:=(1−f−1)V2 gives a direct sum decomposition T M=V3+V4which defines another almost para-complex structure I which anticommutes with K.Hence,(J=IK,K) is an almost hypercomplex structure.It is integrable if and only if the distri-bution V4=(1−f−1)V2associated with the3-web(V1,V2,V3)is integrable. So,any3-web defines an almost para-hypercomplex structure which is para-hypercomplex structure if the distribution V4is integrable.
The monograph[1]contains a detailed exposition of the theory of three-webs, which was started by Bol,Chern and Blaschke and continued by M.A.Akivis and his school.Relations with the theories of G-structures,in particular Grassmann structures,symmetric spaces,algebraic geometry,nomography, quasigroups,non-associative algebras and differential equations of hydrody-namic type are discussed.
2.3.Para-quaternionic structures.An almost para-quaternionic structure on a2n-dimensional manifold M is defined by a3-dimensional subbundle Q of the bundle End(T M)of endomorphisms,which is lo-cally generated by an almost para-hypercomplex structure(I,J,K),i.e. Q x=R I x+R J x+R K x where x∈U and U⊂M is a domain where I,J,K are defined.If Q is invariant under a torsion free connection∇(called a para-quaternionic connection)then Q is called a para-quaternionic structure.The normalizer of the Lie algebra sp1(R)=span(I x,J x,K x)in GL(T x M)is isomorphic to Sp1(R)·GL n(R).So an almost para-quaternionic structure can be considered as a Sp1(R)·GL n(R)-structure and a para-quaternionic structure corresponds to the case when this G-structure is1-flat.Any para-hypercomplex structure(I,J,K)defines a para-quaternionic structure Q=span(I,J,K),since the Obata connection∇is a torsion free connection which preserves Q.The converse claim is not true even locally.
A para-quaternionic structure is generated by a para-hypercomplex struc-ture if and only if it admits a para-quaternionic connection with holonomy
8 D.V.ALEKSEEVSKY,C.MEDORI AND A.TOMASSINI
group Hol⊂GL n(R).
Let T M=H⊗E be an almost Grassmann structure of type(2,n), that is a decomposition of the tangent bundle of a manifold M into a tensor product of a2-dimensional vector bundle H and an n-dimensional bundle E.A non-degenerate2-formωH in the bundle H defines an almost para-quaternionic structure Q in M as follows.For any symplectic basis(h−,h+) of afibre H x we define I x=I H⊗1,J x=J H⊗1,K x=K H⊗1where I H,J H,K H are endomorphisms of H x which in the bases h−,h+are repre-sented by the matrices(1).Then,for x∈M,Q x is spanned by I x,J x,K x.
2.4.Almost para-Hermitian and para-Hermitian structures.Like in the complex case,combining an(almost)para-complex structure K with a ”compatible”pseudo-Riemannian metric g we get an interesting class of geo-metric structures.The natural compatibility condition is the Hermitian con-dition which means that that the endomorphism K is skew-symmetric with respect to g.A pair(g,K)which consists of a pseudo-Riemannian metric g and a skew-symmetric(almost)para-complex structure K is called an(al-most)para-Hermitian structure.An almost para-Hermitian structure on a2n-dimensional manifold can be identified with a GL n(R)-structure. The para-Hermitian metric g has necessary the neutral signature(n,n).A para-Hermitian manifold(M,g,K)admits a unique connection(called the para-Bismut connection)which preserves g,K and has a skew-symmetric torsion.
The K¨a hler formω:=g◦K of an almost para-Hermitian manifold is not necessary closed.
In[50],the authors describe a decomposition of the space of(0,3)-tensors with the symmetry of covariant derivative∇gω(where∇g is the Levi-Civita connection of g)into irreducible subspaces with respect to the natural ac-tion of the structure group GL n(R).It gives an important classification of possible types of almost para-Hermitian structures which is an analogue of Gray-Hervella classification of Hermitian structures.Such special classes of almost para-Hermitian manifolds are almost para-K¨a hler manifolds (the K¨a hler formωis closed),nearly para-K¨a hler manifolds(∇gωis a 3-form)and para-K¨a hler manifolds(∇gω=0).
Almost para-Hermitian and almost para-K¨a hler structures naturally arise on the cotangent bundle T∗M of a pseudo-Riemannian manifold(M,g).The
Levi-Civita connection defines a decomposition Tξ(T∗M)=T vert
ξ(T∗M)+
Hξof the tangent bundle into vertical and horizontal subbundles.This gives
an almost para-complex structure and the natural identification T vert
ξM=
T∗x M=T x M=Hξ,whereξ∈T∗x M and allows to define an almost complex structure and a compatible metric on T∗M.The properties of such para-Hermitian structures on the cotangent bundle are studied in[91].In[28], the authors study almost para-Hermitian manifolds with pointwise constant para-holomorphic sectional curvature,which is defined as in the Hermitian case.They characterize these manifolds in terms of the curvature tensor
HOMOGENEOUS PARA-K¨AHLER EINSTEIN MANIFOLDS9 and prove that Schur lemma is not valid,in general,for an almost para-Hermitian manifold.
A(1,2)-tensorfield T on an almost para-Hermitian manifold(M,g,K)is called a homogeneous structure if the connection∇:=∇g−T preserves the tensors g,K,T and the curvature tensor R of the metric g.In[51],the authors characterize reductive homogeneous manifolds with invariant almost para-Hermitian structure in terms of homogeneous structure T and give a classification of possible types of such tensors T.
Left-invariant para-Hermitian structures on semidirect and twisted products of Lie groups had been constructed and studied in the papers[87],[88],[89], [90].
CR-submanifolds of almost para-Hermitian manifolds are studied in[46]. Four dimensional compact almost para-Hermitian manifolds are considered in[81].The author decomposes these manifolds into three families and es-tablishes some relations between Euler characteristic and Hirzebruch indices of such manifolds.
Harmonic maps between almost para-Hermitian manifolds are considered in[20].In particular,the authors show that a map f:M→N between Riemannian manifolds is totally geodesic if and only if the induced map d f:T M→T N of the tangent bundles equipped with the canonical almost para-Hermitian structure is para-holomorphic.
In[74]it is proved that the symplectic reduction of an almost para-K¨a hler manifold(M,g,K)under the action of a symmetry group G which admits a momentum mapµ:M→g∗provides an almost para-K¨a hler structure on the reduced symplectic manifoldµ−1(0)/G.
An almost para-K¨a hler manifold(M,g,K)can be described in terms of symplectic geometry as a symplectic manifold(M,ω=g◦K)with a bi-Lagrangian splitting T M=T+⊕T−,i.e.a decomposition of the tangent bundle into a direct sum of two Lagrangian(in general non-integrable)dis-tributions T±.An almost para-K¨a hler manifold has a canonical symplectic connection∇which preserves the distributions T±,defined by
∇X±Y±=1
10 D.V.ALEKSEEVSKY,C.MEDORI AND A.TOMASSINI
easily check that for symplectic commuting vectorfields X+,Y+∈Γ(T+) and any Z∈Γ(T−),the following holds
R(X+,Y+)Z=∇X+[Y+,Z]T−−∇Y+[X+,Z]T−
=[X+,[Y+,Z]T−]T−−[Y+,[X+,Z]T−]T−
=[X+,[Y+,Z]]T−−[Y+,[X+,Z]]T−=0,
which shows that R(T+,T+)=0.Here X T−is the projection of X∈T M onto T−.
Let f1,...,f n be independent functions which are constant on leaves of T+.Then the leaves of T+are level sets f1=c1,...,f n=c n and the Hamiltonian vectorfields X i:=ω−1◦d f i commute and form a basis of tangent parallelfields along any leaf L.From the formula for the Poisson bracket it follows
{f i,f j}=ω−1(d f i,d f j)=0=d f i(X j)=X j·f i.
By a classical theorem of A.Weinstein[103]any Lagrangian foliation T+is locally equivalent to the cotangent bundlefibration T∗N→N.More pre-cisely,let N be a Lagrangian submanifold of the symplectic manifold(M,ω) transversal to the leaves of an integrable Lagrangian distribution T+.Then there is an open neighborhood M′of N in M such that(M′,ω|M′,T+|M′) is equivalent to a neighborhood V of the zero section Z N⊂T∗N equipped with the standard symplectic structureωst of T∗N and the Lagrangian fo-liation induced by T∗N→N.In[100]a condition in order that(M,ω,T+) is globally equivalent to the cotangent bundle T∗N,ωst is given.
Let(T∗N,ωst)be the cotangent bundle of a manifold N with the standard symplectic structureωst and the standard integrable Lagrangian distribution T+defined by the projection T∗N→N.The graphΓξ=(x,ξx)of closed 1-formsξ∈Λ1(N)are Lagrangian submanifolds in(T∗N,ωst)transversal to T+.The horizontal distribution T−=H∇⊂T(T∗N)of a torsion free linear connection∇in N is a Lagrangian distribution complementary to T+ [85].Hence any torsion free connection defines an almost K¨a hler structure on(ωst,T+,T−=H∇).Note that the distribution T+is integrable and the distribution T−=H∇is integrable only if the connection∇isflat.We called such a structure a half integrable almost K¨a hler structure.An application of such structures to construction of Lax pairs in Lagrangian dynamics is given in[30].
In[67],the authors define several canonical para-Hermitian connections on an almost para-Hermitian manifold and use them to study properties of 4-dimensional para-Hermitian and6-dimensional nearly para-K¨a hler mani-folds.In particular,they show that a nearly K¨a hler6-manifold is Einstein and a priori may be Ricciflat and that the Nijenhuis tensor N K is parallel
HOMOGENEOUS PARA-K¨AHLER EINSTEIN MANIFOLDS11
with respect to the canonical connection.They also prove that the Kodaira-Thurston surface and the Inoue surface admit a hyper-paracomplex struc-ture.The corresponding para-hyperHermitian structures are locally(but not globally)conformally para-hyperK¨a hler.
2.5.Para-K¨a hler(bi-Lagrangian or Lagrangian2-web)structures.
A survey of results on geometry of para-K¨a hler manifolds is given in[47]. Here we review mostly results which are not covered in this paper.Re-call that an almost para-Hermitian manifold(M,g,K)is para-K¨a hler if the Levi-Civita connection∇g preserves K or equivalently its holonomy group Hol x at a point x∈M preserves the eigenspaces decomposition T x M=T+x+T−x.The parallel eigendistributions T±of K are g-isotropic integrable distributions.Moreover,they are Lagrangian distributions with respect to the K¨a hler formω=g◦K which is parallel and,hence,closed. The leaves of these distributions are totally geodesic submanifolds and they areflat with respect to the induced connection,see section2.4.
A pseudo-Riemannian manifold(M,g)admits a para-K¨a hler structure (g,K)if the holonomy group Hol x at a point x preserves two complemen-tary g-isotropic subspaces T±x.Indeed the associated distributions T±are parallel and define the para-complex structure K with K|T±=±1.
Let(M,g)be a pseudo-Riemannian manifold.In general,if the holonomy group Hol x preserves two complementary invariant subspaces V1,V2,then a theorem by L.Berard-Bergery[76]shows that there are two complemen-tary invariant g-isotropic subspaces T±x which define a para-K¨a hler structure (g,K)on M.
Many local results of K¨a hler geometry remain valid for para-K¨a hler mani-folds,see section4.The curvature tensor of a para-K¨a hler manifold belongs to the space R(gl n(R))of gl n(R)-valued2-forms which satisfy the Bianchi identity.This space decomposes into the sum of three irreducible gl n(R)-invariant subspaces.In particular,the curvature tensor R of a para-K¨a hler manifold has the canonical decomposition
R=cR1+R ric0+W
where R1is the curvature tensor of constant para-holomorphic curvature 1(defined as the sectional curvature in the para-holomorphic direction (X,KX)),R ric0is the tensor associated with the trace free part ric0of the Ricci tensor ric of R and W∈R(sl n(R))is the para-Weil tensor which has zero Ricci part.Para-K¨a hler manifolds with constant para-holomorphic curvature(and the curvature tensor proportional to R1)are called para-K¨a hler space forms.They were defined in[50]and studied in[95],[44], [45].Like in the K¨a hler case,there exists(up to isometry)a unique simply connected complete para-K¨a hler manifold M k of constant para-holomorphic curvature k.It can be described as the projective space over para-complex numbers.Any complete para-K¨a hler manifold of constant para-holomorphic curvature k is a quotient of M k by a discrete group of isometries acting in a。